$$
\begin{array}{l}
\lim \limits_{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sin \left(t^{2}\right) d t}{x^{6}} \quad \frac{0}{0} \text { 型 } \\
\begin{array}{r}=\lim \limits_{x \rightarrow 0} \frac{\sin x^{4} \cdot 2 x}{6 x^{5}} \quad \text { 此步的求导流程 } \varphi(x)=\int_{0}^{x^{2}} \sin \left(t^{2}\right) d t \\
\varphi^{\prime}(x)=\sin \left(\left(x^{2}\right)^{2}\right) \cdot\left(x^{2}\right)^{\prime}=\sin x^{4} \cdot 2 x\end{array} \\
=\lim \limits_{x \rightarrow 0} \frac{\sin x^{4}}{3 x^{4}} \\
\end{array}
$$